Solve simultaneous equations using the inverse matrix

Columns follow one variable order throughout

For 2x+y=7 and x+3y=11, use the column vector (x,y)ᵀ. Each coefficient row must follow x then y, including a zero when a variable is absent

Given matrix

\(\displaystyle A=\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}\)

Find the determinant of this coefficient matrix

Keep exact values until the interpretation stage

The solution vector is A⁻¹b. The inverse and multiplication are displayed. Substituting x=2 and y=3 verifies both original equations

Given matrix

\(\displaystyle \begin{pmatrix}2&1\\1&3\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}7\\11\end{pmatrix}\)

What is y in the displayed system?

Which technology workflow checks a computed inverse solution?

Singular systems need a consistency test

The pair x+2y=5 and 2x+4y=10 has infinitely many solutions: x=5−2t,y=t for real t. Changing the second right-hand side to 11 makes it inconsistent. Both coefficient matrices have determinant zero

Analyse singular alternatives, ungraded

Explain why det(A)=0 alone cannot distinguish no solutions from infinitely many. Give a complete parameterisation for the consistent example and prove inconsistency for the changed example. Independent written analysis is ungraded

Model: compare whole equations

The second left-hand side is exactly twice the first. With right-hand side 10 it adds no constraint, so any real y=t gives x=5−2t and satisfies both equations; conversely every solution has this form. With right-hand side 11, doubling the first equation would require 2x+4y=10 and 11 simultaneously, a contradiction